Wikipedia / Order embedding
Wikipedia / Order embedding
In order theory, a branch of mathematics, an order embedding is a special kind of monotone function, which provides a way to include one partially ordered set into another. Like Galois connections, order embeddings constitute a notion which is strictly weaker than the concept of an order isomorphism. Both of these weakenings may be understood in terms of category theory.
Formal definition
, a function is an order embedding is both order-preserving and order-reflecting, one has
Such a function is necessarily injective..
Properties
An order isomorphism can be characterized as a surjective order embedding. As a consequence, any order embedding f restricts to an isomorphism between its domain S and its image f(S), which justifies the term "embedding". On the other hand, it might well be that two (necessarily infinite) posets are mutually order-embeddable into each other without being order-isomorphic.
An example is provided by the open interval of real numbers and the corresponding closed interval maps the former to the subset is both order-preserving and order-reflecting (because it is an affine function has a least element does not. For a similar example using arctan to order-embed the real numbers into an interval, and the identity map for the reverse direction, see e.g. Just and Weese (1996).
of order-preserving maps whose composition is called a coretraction, and must be an order embedding. of divisors of 6, partially ordered by x divides y, but there is no such place.
Additional perspectives
Posets can straightforwardly be viewed from many perspectives, and order embeddings are basic enough that they tend to be visible from everywhere. For example:
(Model theoretically) A poset is a set equipped with a (reflexive, antisymmetric and transitive) binary relation. An order embedding A → B is an isomorphism from A to an elementary substructure of B.
(Graph theoretically) A poset is a (transitive, acyclic, directed, reflexive) graph. An order embedding A → B is a graph isomorphism from A to an induced subgraph of B.
(Category theoretically) A poset is a (small, thin, and skeletal) category such that each homset has at most one element. An order embedding A → B is a full and faithful functor from A to B which is injective on objects, or equivalently an isomorphism from A to a full subcategory of B.